Question: You have found the following ages (in years) of 4 sloths. Those sloths were randomly selected from the 28 sloths at your local zoo: $ 12,\enspace 12,\enspace 6,\enspace 1$ Based on your sample, what is the average age of the sloths? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 28 sloths, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\overline{x}} = \dfrac{12 + 12 + 6 + 1}{{4}} = {7.8\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {17.64} + {17.64} + {3.24} + {46.24}} {{4 - 1}} $ {s^2} = \dfrac{{84.76}}{{3}} = {28.25\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{28.25\text{ years}^2}} = {5.3\text{ years}} $ We can estimate that the average sloth at the zoo is 7.8 years old. There is also a standard deviation of 5.3 years.